# Embedding of a Blade-Element Analytical Model into the SHYFEM Marine Circulation Code to Predict the Performance of Cross-Flow Turbines

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Preliminary Validation of the ANSYS CFD Model

_{P}, which is defined as:

_{P}-TSR describes the turbine performance as a function of the Tip Speed Ratio, TSR, which is defined as:

_{P}predicted from 2D simulations is overestimated since important 3D effects—first of all, the blade tip losses—cannot be taken into account. In particular, due to the moderate value of the chord-based aspect ratio of this turbine (blade length/chord = 7.5), significant tip losses are expected. Since these 2D CFD simulations cannot directly predict tip losses, a correction factor must be used to reduce C

_{P}afterwards in order to allow a meaningful comparison with experimental data. On the basis of the study [41], a plausible value of tip losses for a CFT with this aspect ratio could be ≈25%. Figure 1 depicts the results of CFD (having already subtracted 25% from the 2D original predictions) compared to the experimental results obtained for a flow speed of 10 m/s [40]. It can be seen that the trend, the optimal TSR, and the maximum C

_{P}are well matched, with just a small discrepancy in the optimal TSR. In order to match the experimental data at the low TSRs, particular attention was paid to the quality of the calculation grid (Figure 2); in fact, around each airfoil, many layers of regular quadrilateral cells were created of increasing height with exponential law starting from the blade wall. The following parameters have been kept for all the 2D grids with blades adopted in this study: 710 cells around the blade profile; c × 4 × 10

^{−6}(where c is the airfoil chord) for the height of the first layer of cells at the blade wall; 1.06 for the cell height growing ratio; 110 layers around the blade profile. It should be observed that the cell smallest height depends on the particular chord value. In this way, around each blade, a very fine and high-quality grid is obtained for a distance of about 1.5 × c from the airfoil/hydrofoil.

#### 2.2. BEM Theory Basics

_{L}and C

_{D}are the lift and drag coefficients of the particular hydrofoil that is considered. C

_{L}and C

_{D}depend on the attack angle and Reynolds number, and they need to be measured experimentally in a wind tunnel. Then, the expressions of torque $\delta Q$ and power $\delta P$ are

#### 2.3. Grid Effect Analysis With the ANSYS Hybrid Model

_{P}of the single blade during one revolution; the hybrid model is sufficiently able to capture the main detail of the C

_{P}evolution at the different grid resolutions except for the coarsest case (i.e., 20 cells on the ring). The anomalous and unphysical drop in the upwind curve (θ ≈ 95°–110°) is due to the virtual camber sub-model. In fact, this sub-model is greatly sensitive to the turbine solidity: for solidity lower than 4–5% it is stable, but it does not adequately work for the solidity of the Kobold turbine, therefore leading to the choice to switch off this sub-model.

_{P}calculated, in case of 44 cells on the ring, with the hybrid model and with pure CFD for several values of the turbine TSR.

#### 2.4. SHYFEM Grid

#### 2.5. Friction’s Formulation in the SHYFEM Hybrid Model

## 3. Results of the BEM–SHYFEM Hybrid Model

#### 3.1. Behavior of the Single Turbine at Different TSR

_{P}sufficiently matches the respective CFD result for all the TSR, but also the behavior of blade instantaneous C

_{P}satisfies the typical azimuthal evolution in CFTs, showing, for instance, dynamic stall in upwind at very low TSR and power output decreasing in downwind as the TSR increases. What emerges is a quantitative difference between the two hybrid models and the pure CFD: as explained before, this is due to the fact that the model developed in [35] is tuned for an airfoil with lower solidity. We report also the ANSYS hybrid model in graphics just to make a comparison between the two hybrid models. Since both are based on the same analytical model, the same behavior is expected. As shown in Figure 13, expectations are satisfied for all TSR. By comparing the values of the source terms in ANSYS and SHYFEM simulations (Table 3), it is evident that the total source terms are similar: differences are limited to 10% (the only exception is TSR 3.2).

#### 3.2. Application to a Small Turbine’s Cluster

_{P}curve than in the isolated case, as confirmed in Figure 17. For turbines 2 and 3, acceleration corridors are present respectively at the beginning and at the end of upwind path. In fact, in Figure 17, it is shown that turbine 2 anticipates the production, while turbine 3 postpones production with respect to the isolated turbine. The behavior of the three turbines is in accordance with pure CFD results. Turbines 2 and 3 have clockwise rotational verse: in this case, the θ reference system is not the same as that proposed in Figure 5a but it is opposite with respect to the y axis. The direction of increase of the angle θ is concordant with the direction of rotation.

## 4. Conclusions

_{P}of the blade during a turbine revolution. This correspond to cell sizes of about 1/17 of the turbine diameter. The SHYFEM hybrid model matches the results of the pure CFD and ANSYS hybrid model also in a wide range of TSR values.

_{P}and the turbine effects on the surrounding and on the downstream flows depends on grid spacing.

_{P}increasing/decreasing with respect to the isolated turbine, but also regarding the local mutual interactions between the devices. This is not trivial, given the highly asymmetrical behavior of CFTs with respect to the plain passing through the turbine axis and parallel to the current.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**ANSYS-Fluent CFD C

_{P}-TSR curve vs. experimental data, concerning tests in a wind tunnel performed at a flow speed of 10 m/s. TSR: Tip Speed Ratio.

**Figure 4.**(

**a**) Sketch of the Kobold turbine; (

**b**) detail of the ANSYS grid with 44 elements on the ring.

**Figure 5.**(

**a**) Scheme of upwind and downwind path; (

**b**) C

_{P}for a single blade at different resolution.

**Figure 6.**Wake velocity profile at 2R, (

**a**) x velocity and (

**b**) y velocity and at 10R behind the turbine (

**c**) x velocity and (

**d**) y velocity.

**Figure 8.**Qualitative equivalence between (

**a**) CFD symmetrical airfoil and hybrid model with virtual camber and (

**b**) CFD cambered airfoil and hybrid model without virtual camber.

**Figure 10.**C

_{P}comparison between pure CFD and SHYFEM hybrid model with and without dynamic stall model in downwind.

**Figure 11.**Qualitative scheme for assigning θ value (

**a**) to upwind inside elements and (

**b**) to downwind inside elements.

**Figure 13.**C

_{P}comparison between pure CFD, ANSYS, and SHYFEM hybrid models at different TSR values 1.2, 1.7, 2.3, 2.7, 2.85, 3.2, 4 respectively in (

**a**–

**g**) and the averaged C

_{P}in figure (

**h**).

**Figure 14.**Qualitative flow field detail for (

**a**) the ANSYS hybrid model and (

**b**) the SHYFEM hybrid model.

**Figure 15.**Arrangement of turbines in the two considered configurations (top view); the flow is flowing from left.

**Figure 16.**Flow field comparison between the SHYFEM hybrid model, at two different grid resolutions, and pure CFD (top view).

Turbulence Model | SST k-ω | |
---|---|---|

Solution Methods | Pressure–Velocity Coupling | Scheme SIMPLEC |

Spatial Discretization | Gradient | Least Squares Cell-Based |

Pressure | 2nd Order | |

Momentum | 2nd-Order Upwind | |

TKE | 2nd-Order Upwind | |

Specific Dissipation Rate | 2nd-Order Upwind | |

Transient Formulation | 2nd-Order Implicit | |

Transient Formulation | Second-Order Implicit | |

Residuals | 0.0001 |

Time Step (idt) | 1 s |

End Simulation Time (itend) | 1500 s |

Flow | 1680 m^{3}/s |

Friction | Ireib 5 czdef 0.02 |

External Forces | NO |

(N/m^{3}) | Sources x | Sources y | Total Sources | C_{P} Averaged | |
---|---|---|---|---|---|

TSR 1,2 | UDF | 16˙004 | 22˙582 | 38˙586 | 0.16 |

SHYFEM | 35˙722 | 1˙520 | 37˙242 | 0.19 | |

TSR 1,7 | UDF | 23˙063 | 33˙330 | 56˙393 | 0.28 |

SHYFEM | 49˙795 | 3˙425 | 53˙220 | 0.29 | |

TSR 2,3 | UDF | 29˙681 | 42˙803 | 72˙484 | 0.40 |

SHYFEM | 65˙200 | 7˙292 | 72˙492 | 0.40 | |

TSR 2,7 | UDF | 46˙897 | 33˙271 | 80˙168 | 0.45 |

SHYFEM | 74˙618 | 11˙518 | 86˙136 | 0.43 | |

TSR 2,85 | UDF | 35˙425 | 46˙865 | 82˙290 | 0.44 |

SHYFEM | 77˙248 | 12˙946 | 90˙194 | 0.44 | |

TSR 3,2 | UDF | 36˙857 | 48˙781 | 85˙638 | 0.42 |

SHYFEM | 84˙372 | 17˙072 | 101˙444 | 0.40 | |

TSR 4 | UDF | 39˙079 | 51˙375 | 90˙454 | 0.33 |

SHYFEM | 83˙242 | 18˙737 | 101˙979 | 0.33 |

% | SHYFEM Hybrid | Pure CFD |
---|---|---|

Turbine 1 (side by side) | 17.75 | 17.53 |

Turbine 1 (triangular) | 0.78 | −5.97 |

Turbine 2 | 25.62 | 19.76 |

Turbine 3 | 28.81 | 18.37 |

Turbine 4 | 21.93 | 20.71 |

Turbine 5 | 24.95 | 19.37 |

Side by side triad | 24.06 | 18.55 |

Triangular triad | 15.89 | 11.37 |

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**MDPI and ACS Style**

Pucci, M.; Bellafiore, D.; Zanforlin, S.; Rocchio, B.; Umgiesser, G.
Embedding of a Blade-Element Analytical Model into the SHYFEM Marine Circulation Code to Predict the Performance of Cross-Flow Turbines. *J. Mar. Sci. Eng.* **2020**, *8*, 1010.
https://doi.org/10.3390/jmse8121010

**AMA Style**

Pucci M, Bellafiore D, Zanforlin S, Rocchio B, Umgiesser G.
Embedding of a Blade-Element Analytical Model into the SHYFEM Marine Circulation Code to Predict the Performance of Cross-Flow Turbines. *Journal of Marine Science and Engineering*. 2020; 8(12):1010.
https://doi.org/10.3390/jmse8121010

**Chicago/Turabian Style**

Pucci, Micol, Debora Bellafiore, Stefania Zanforlin, Benedetto Rocchio, and Georg Umgiesser.
2020. "Embedding of a Blade-Element Analytical Model into the SHYFEM Marine Circulation Code to Predict the Performance of Cross-Flow Turbines" *Journal of Marine Science and Engineering* 8, no. 12: 1010.
https://doi.org/10.3390/jmse8121010